Technical Note—On Dynamic Pricing with Covariates

We consider dynamic pricing with covariates under a generalized linear demand model: A seller can dynamically adjust the price of a product over a horizon of T time periods, and at each time period t, the demand of the product is jointly determined by the price and an observable covariate vector $x_t\in {\mathbb{R}}^d$ through a generalized linear model with unknown coefficients. Most of the existing literature assumes the covariate vectors x_ts are independently and identically distributed (i.i.d.); the few papers that relax this assumption either sacrifice model generality or yield suboptimal regret bounds. In this paper, we show that Upper Confidence Bound and Thompson sampling-based pricing algorithms can achieve an $O(d\sqrt{T}\log T)$ regret upper bound without assuming any statistical structure on the covariates x_t. Our upper bound on the regret matches the lower bound up to logarithmic factors. We thus show that (i) the i.i.d. assumption is not necessary for obtaining low regret, and (ii) the regret bound can be independent of the (inverse) minimum eigenvalue of the covariance matrix of the x_ts, a quantity present in previous bounds. Moreover, we consider a constrained setting of the dynamic pricing problem where there is a limited and unreplenishable inventory, and we develop theoretical results that relate the best achievable algorithm performance to a variation measure with respect to the temporal distribution shift of the covariates. We also demonstrate the proposed algorithms’ performance with numerical experiments.

Supplemental Material: All supplemental materials, including the code, data, and files required to reproduce the results, are available at https://doi.org/10.1287/opre.2021.0802.